Fair Clustering Through Fairlets

We study the question of fair clustering under the {\em disparate impact} doctrine, where each protected class must have approximately equal representation in every cluster. We formulate the fair clustering problem under both the $k$-center and the $k$-median objectives, and show that even with two protected classes the problem is challenging, as the optimum solution can violate common conventions---for instance a point may no longer be assigned to its nearest cluster center! En route we introduce the concept of fairlets, which are minimal sets that satisfy fair representation while approximately preserving the clustering objective. We show that any fair clustering problem can be decomposed into first finding good fairlets, and then using existing machinery for traditional clustering algorithms. While finding good fairlets can be NP-hard, we proceed to obtain efficient approximation algorithms based on minimum cost flow. We empirically quantify the value of fair clustering on real-world datasets with sensitive attributes

Improved Differentially Private Densest Subgraph: Local and Purely Additive

We study the Densest Subgraph problem under the additional constraint of differential privacy. In the LEDP (local edge differential privacy) model, introduced recently by Dhulipala et al. [FOCS 2022], we give an $(\epsilon, \delta)$-differentially private algorithm with no multiplicative loss: the loss is purely additive. This is in contrast to every previous private algorithm for densest subgraph (local or centralized), all of which incur some multiplicative loss as well as some additive loss. Moreover, our additive loss matches the best-known previous additive loss (in any version of differential privacy) when $1/\delta$ is at least polynomial in $n$, and in the centralized setting we can strengthen our result to provide better than the best-known additive loss. Additionally, we give a different algorithm that is $\epsilon$-differentially private in the LEDP model which achieves a multiplicative ratio arbitrarily close to $2$, along with an additional additive factor. This improves over the previous multiplicative $4$-approximation in the LEDP model. Finally, we conclude with extensions of our techniques to both the node-weighted and the directed versions of the problem.Comment: 41 page